  Restricted Flow and Orifice Studies

 In order to build a computer model of the Levitator, and thereby further validate the theory of Radial Momentum, I conduct various tests to determine formulas for the basic behavior of air as it flows through a central orifice and out between the table and the plate. In addition to determining the behavior of air, these tests also provide practice calibrating and using the flow and pressure instrumentation. In these studies, I measure the mass flux rate (kg/sec) that escapes through a round orifice in a plenum, as a function of the pressure in the plenum. The flux turns out to be proportional to the square root of pressure for escape velocities less than the speed of sound. For higher velocities, it increases directly with pressure. Schematic of device to measure flow through a plate valve. Air from the plenum at the left at [P, T] flows through the round orifice with diameter [D] and then through a gap between the plates of width [h]. The bottleneck valve is a ring of diameter [D] and height [h]. The question mark [?] indicates the flux, to be determined. In these trials D = 1/8".

 The Device to Measure Flux as a Function of Pressure Air flows through the flow meter (left), past the pressure meters, into a plenum above the test table and out through the orifice in the table. I use two different tables, with orifice sizes of 1/16" and 1/8". I vary the plenum pressure by adjusting the pump (under the flow meter and not shown). I measure the pressure and mass flux with the gauges. The function turns out to have two different forms, depending on the velocity of flow through the orifice.

 Detail of Plenum and Orifice View from Under the table Air enters the plenum through the hose from the pressure gauge and exits through the orifice in the test table. The plenum is a cylinder about an inch long and 1/4" in diameter. As the pressure varies from zero to about 60 psig the mass flux through the orifice increases. The device is made from smooth acrylic.

 Schematic of Plenum and Orifice In this schematic, air enters the plenum from the left, with pressure [P] and temperature [T]. I assume velocity and momentum in the plenum to be negligible. The air evacuates through the round orifice with diameter [D] into ambient condition [P0, T0]. I use two tables, one with D = 1/8" and one with D = 1/16". The question mark [?] represents the unknown, which in this case is the mass flux (kg/sec).

 As a continuation of the basic tests on orifices, I place a plate at various distances from the orifice as a valve and measure the flux versus pressure. Except at low pressures, the flux turns out to be monotonic with the orifice area. The Basic Result As the pressure in the plenum increases, it pushes air through the orifice at a higher rate so the mass flux rate (kg/sec) increases. The mass flux through at 1/8" is substantially higher than the mass flux through at 1/16". The mass flux for both orifices rises quickly at lower pressures, then less rapidly, indicating a change in function. Flux Proportional to Area This chart shows the ratio of the flux at 1/8" versus the flux at 1/16" at various pressures. This ratio stays in proportion at about 4:1. This is also the ratio of the areas of the two orifices. This indicates flux is proportional to area.

 In order to derive reasonable formulas to plug in to the simulation model to represent the central valve process, I use basic physics, logic and curve fitting. The goal at this point is not to re-invent orifice flow physics so much as to derive reasonably useful formulas. For low pressures, the air flows gently and smoothly through the orifice. From basic energy balance, the energy loss from the pressure drop through the orifice accelerates the air to a higher kinetic energy level. Since kinetic energy = 1/2 mv2, the velocity, then, is proportional to the square root of the pressure drop. Mass flux is velocity * density * cross section. The effective orifice is somewhat smaller than the actual orifice since the airflow bends and narrows in the orifice to create a "vena contracta" effect of about 85% efficiency. The combined equation for flux, then, is: Flux = sqrt(2 * pressure_drop / density) * orifice_area * exit_density * 0.85 As the air exit velocity reaches the speed of sound, the relationship between pressure and flux changes. Air does not generally flow in open systems at speeds greater than the speed of sound. It compacts and forms a shock wave. As the pressure increases, the exit air speed remains at 343 meters / second while the density of the air squeezing through the orifice continues to increase. Therefore, the flux continues to increase directly with the pressure, per: Flux = orifice_area * exit_density * speed_of_sound * ( 1 + pressure_drop / p0) * .62 Where p0 is about 100 k-pa. That and a 62% vena contracta coefficient for compacted flow reconciles the data. The combination of these two equations seems to predict the data fairly well over the entire range of observations. Again, the goal here is not to develop defensible orifice physics so much as to derive reasonable formulas to plug into the simulation model. Two Formulas for Flux The actual data is shown in blue. At pressures below 100 k-pa, flux is proportional to the square root of pressure (equation shown in green). At higher pressures, it increases directly with pressure (equation shown in red). The Fit Another way to view the fit is to plot the model flux against the actual flux. A perfect fit would be a straight line through equal values. The combination of the two formulas appears to provide a fairly good fit for most of the data.

The Data

 Column Description of Columns Pressure Gauge psig (pounds per square inch, gauge) - reading directly from the pressure gauge. Pressure Gauge k-pa (kilo-pascals) - gauge pressure psig times 6895 k-pa/psi. Density Density - 1.2 kg/m3 times (11.6/14.6 + gauge k-pa / 78.91). At Lake Tahoe, the location of the tests, the ambient pressure is 11.6 psi or 78.91 k-pa. At sea level, the presure is 14.6 psi and the density of air is 1.2 kg/m3 . Flux-1/16 1/16" diameter numerical reading directly from flow meter for 1/16" diameter orifice. Flux-1/8 1/8" - same as above. CF Conversion Factor - from MEM Flow Products Company, the flow meter manufacturer. The factor adjusts gauge readings for different ambient temperature and pressure. The meter is calibrated for base conditions of 70 degrees F and 100 psig. I ran the tests at 50 degrees F and 11.6 psig (Lake Tahoe). The formula for the factor is: Where Pg is the operating pressure + 11.6 psi, Ps is the base pressure (100 psi) + 11.6 psi, Ts is the base temperature (70F) + 460F and Tg is the operating Temperature, (50F) + 460F. Example, the factor for 5 psig at 50F is f = sqrt(16.6/111.6 * 530/510) = .39316. Flux-1/16 1/16" Base Flux (scfm) - gauge reading times adjuster gives the flow at base conditions (70 deg. F and 14.6 psi) Flux-1/8 1/8" - same as above. Flux-1/16 1/16" Flux (kg/sec) - base flux (scfm) times 4.5 nt/lb times .07849 lb/scfm / 9.81 m/s2 / 60 s/min converts scfm to kg/sec. Flux-1/8 1/8" - same as above

 Pressure Pressure Density Flux-1/16 Flux-1/8 CF Flux-1/16 Flux-1/8 Flux-1/16 Flux-1/8 gauge k-pa kg/m^3 gauge gauge factor scfm scfm kg/s kg/s 0.5 3.4 1.0 2.50 0.3357 0.8392 4.75E-04 1.0 6.9 1.1 0.80 3.60 0.3425 0.2740 1.2331 1.55E-04 6.98E-04 1.5 10.3 1.1 4.40 0.3493 1.5368 8.70E-04 2.0 13.8 1.2 1.20 5.00 0.3559 0.4270 1.7793 2.42E-04 1.01E-03 2.5 17.2 1.2 5.50 0.3624 1.9929 1.13E-03 3.0 20.7 1.3 1.60 6.00 0.3687 0.5900 2.2123 3.34E-04 1.25E-03 3.5 24.1 1.3 6.40 0.3750 2.3999 1.36E-03 4.0 27.6 1.4 1.80 6.70 0.3811 0.6860 2.5536 3.89E-04 1.45E-03 4.5 31.0 1.4 7.10 0.3872 2.7491 1.56E-03 5.0 34.5 1.5 2.00 7.40 0.3932 0.7863 2.9094 4.45E-04 1.65E-03 5.5 37.9 1.5 7.70 0.3990 3.0726 1.74E-03 6.0 41.4 1.6 2.10 8.10 0.4048 0.8502 3.2792 4.82E-04 1.86E-03 6.5 44.8 1.6 8.30 0.4105 3.4075 1.93E-03 7.0 48.3 1.7 2.30 8.60 0.4162 0.9572 3.5791 5.42E-04 2.03E-03 8.0 55.2 1.8 2.40 0.4272 1.0253 5.81E-04 9.0 62.1 1.9 2.50 0.4380 1.0950 6.20E-04 10.0 69.0 2.0 2.60 0.4485 1.1661 6.60E-04 15.0 103.4 2.5 2.90 0.4977 1.4433 8.18E-04 20.0 137.9 3.1 3.10 0.5425 1.6816 9.52E-04 25.0 172.4 3.6 3.30 0.5838 1.9265 1.09E-03 30.0 206.9 4.1 3.45 0.6224 2.1473 1.22E-03 35.0 241.3 4.6 3.70 0.6587 2.4373 1.38E-03 40.0 275.8 5.1 3.85 0.6932 2.6687 1.51E-03 45.0 310.3 5.7 4.05 0.7260 2.9402 1.67E-03 50.0 344.8 6.2 4.20 0.7574 3.1810 1.80E-03 55.0 379.2 6.7 4.35 0.7875 3.4257 1.94E-03 60.0 413.7 7.2 4.50 0.8165 3.6744 2.08E-03 65.0 448.2 7.8 4.63 0.8446 3.9061 2.21E-03